Extremely large magnetoresistance in twisted intertwined graphene spirals

Extremely large magnetoresistance (XMR) is highly applicable in spintronic devices such as magnetic sensors, magnetic memory, and hard drives. Typically, XMR is found in Weyl semimetals characterized by perfect electron–hole symmetry or exceptionally high electric conductivity and mobility. Our study explores this phenomenon in a recently developed graphene moiré system, which demonstrates XMR owing to its topological structure and high-quality crystal formation. We investigate the electronic properties of three-dimensional intertwined twisted graphene spirals (TGS), manipulating the screw dislocation axis to achieve a rotation angle of 7.3°. Notably, at 14 T and 2 K, the magnetoresistance of these structures reaches 1.7 × 107%, accompanied by a metal–insulator transition as the temperature increases. This transition becomes noticeable when the magnetic field exceeds a minimal threshold of approximately 0.1 T. These observations suggest the possible existence of complex, correlated states within the partially filled three-dimensional Landau levels of the 3D TGS system. Our findings open up possibilities for achieving XMR by engineering the topological structure of 2D layered moiré systems.

specifically identified as the a-axis.The application of current density is perpendicular to the a-axis, while the magnetic field aligns parallel to the c-axis, designated as the direction of spiral growth.Electrodes, derived from silver glue, span both sides and surfaces of the sample.The distance between the two vertical voltage terminals measures 0.8 mm, while the separation between the two horizontal terminals is 2.6 mm.

Angular-dependent magnetoresistance.
To further confirm the certain twist angle in long range, we also studied the angulardependent in-plane magnetoresistivity.During the in-plane rotation of the magnetic field, the current was applied along in-plane while the initial field direction (Φ = 0°) is set to be parallel to current as shown in schematic image in Supplementary Figure 2a.The angle-dependent resistance R at 2 K and 9 T reveals the symmetry of the orbital part of the electronics structure which corresponds to the crystal lattice.It is expected that a two-fold symmetry for the in-plane resistance would occur due to the Lorentz force since the current were applied within the ab-plane as well (Supplementary Figure 2b).Nevertheless, besides the background of the two-fold symmetry, there are additional oscillations caused by other contribution.Thus, we subtract the background from the two-fold symmetry component, and observe other periodic signals as can be visualized in the middle and bottom of Supplementary Figure 2b.Surprisingly, the angular dependent ΔR reveals obvious high frequency oscillations.By applying fast Fourier transform (FFT) analysis on the oscillations, four period can be identified as the oscillation period of ΔR as shown in Supplementary Figure 2c, in which the first period T1 denotes a peak at 7.8°, and the following three harmonics periods T2, T3 and T4 are also visible as 15.3°, 23.3°, and 30.7°, respectively, indicating a set of high order period as Tn= nT1.Therefore, the 7.8° oscillation of the angular-dependent in-plane magnetoresistivity is accordance with the twist angle of the GS.The findings presented in our article specifically pertain to GS with a large twist angle.It is worth noting that GS with a small twist angle exhibits numerous peculiar properties, and these are discussed in detail in our related work [1].In relatively low temperatures, MR of the twisted GS reveals obvious SdH oscillations as shown in Figure 3b.Thus, we studied the magnetoresistivity (ρ) comprehensively at 2 K under magnetic fields applied in from directions from out-of-plane to in-plane as shown in Supplementary Figure 5a.Here, the electric current was applied within the ab-plane, and the magnetic field was applied perpendicular to the electric current and rotated out of the ab-plane.θ refers to the angle between the direction of the magnetic Meanwhile, the carrier concentration also follows the 3D model:  3 =       /(3 2 ), where   represents the Fermi wave-vector along the i-direction [6].The discrepancy between the Fermi surface and carrier concentration arises notably due to the weak dispersion along   .

The graphene Chemical Vapor Deposition (CVD) method
The spiral stacking structure of SP2-bonded carbon has been a subject of interest since the 1960s, as evidenced by early work.With the advent of the graphene Chemical Vapor Deposition (CVD) method, numerous research groups have demonstrated the feasibility of producing spiral graphene structures using this standard CVD process [7,8].Remarkably, spiral graphene structures can also be achieved by directly annealing SP2 carbon [9], underscoring the thermodynamic stability of the spiral configuration in graphene.Our recent research has further advanced this field through the development of a graphene origami-kirigami approach.This method involves processes such as wrinkling, folding, tearing, and cracking, leading to the spiral growth of graphene multilayers with controlled stacking orders.The intricacies of this graphene spiral growth process have been elaborated in our latest publication [10], providing a comprehensive understanding of this unique structural phenomenon.In current work, we more focus on the transport properties of the spiral graphene.

Figure 2 Supplementary Figure 3 Supplementary Figure 4
Angle dependent in-plane magnetoresistance.(a) A schematic illustration of the measurement configuration.The electric current is applied within the ab-plane of the single crystal, and the applied magnetic field is rotated within the ab-plane as well.(b) Angular dependence of magnetoresistance measured at 9 T demonstrates a major two-fold symmetry owing to the Lorentz force effect from B×I.The periodic resistance ΔR is also given by subtracting the two-fold background, and such periodic oscillation can be well identified in the enlarged view in bottom.(c) FFT spectra for the periodic resistance ΔR.Peaks of 7.8°, 15.3°, 23.3°, and 30.7° are corresponding to the 1 st , 2 nd , 3 rd and 4 th harmonics of oscillations, which are well consistent with the result of AFM and TEM.Quantum oscillation.(a) ρxx(B) GS data at 2 K shows SdH oscillations.The blue line has shown the B 2 fitting which can be described as a magnetoresistance background.(b) SdH oscillations after subtracting the background from the 2 K ρxx measurements.(c) Peaks e1,2, h1,2 correspond to the 1st and 2nd harmonics of oscillations from electrons and holes.(d) In order to distinguish the signals from electrons and holes, a low-pass filter has been added when subtracting the background.The FFT result of filtered SdH oscillations have been shown in (d).(e, f) SdH oscillations in temperatures from 2 to 10 K. Inset shows the temperature dependence of the relative amplitude of Δρ for the SdH oscillation at 1/B = 0.847 T -1 .The solid line is a fit to the Lifshitz-Kosevich formula:   =  is defined as thermal damping factor, α = 2 2     ℏ ⁄ ≈ 14.69T/K ,   is Boltzmann constant, me is the bare mass of the electron.The result of LK fitting shows the hole effective mass of GS (  * = 0.027 ± 0.00028  ) is smaller than NG (  * = 0.036 ± 0.00129  ).Anomalous metal-insulator transition in twisted graphene spiral.(a)The temperature dependent resistivity of GS at different magnetic field from 0 to 14 T.An unconventional transition from metal to isolator appears when an infinite external magnetic field has been applied.Inset shows the magnetic field dependence of transition temperature T * .(b) The temperature dependent MR at magnetic field ranging from 3T to 14 T, exhibiting an extremely large MR even at 300 K.The magnitude of MR reaches 3.4 × 10 4 % at 300 K, 14 T, and exceed 10 3 % at 300 K, 3 T.

Supplementary Figure 5 Supplementary Figure 6 1 𝑒(
field and the c-axis, as shown inset of Supplementary Figure 5a.When the field is applied away from c-axis, the magnetoresistivity is strongly suppressed from  9T =0° = 122 mΩ • cm to  9T =90° = 1.99 mΩ • cm, indicating the highly anisotropic nature of the carrier transport in the twisted GS.The splitting of the first oscillations (n = 1) of electron and hole can be clearly observed at B > 3 T (arrows in Supplementary Figure 5a), which originate from the conduction-electron factor g-shift in the Fermi energy under magnetic field [2-4].It should be noted that only the perpendicular component of the magnetic field affects the band splitting due to a quasi-2D Fermi surface.Due to the weak desperation along kz direction, one can hardly distinguish between quasi-2D behavior and 2D behavior through SdH oscillations.Quasi-2D Fermi surface.(a) Magnetoresistivity for GS under magnetic field applied from out-of-plane (θ = 0°) to in-plane (θ = 90°).Inset shows a schematic view of magnetoresistivity measurement.The electric current is applied within the in-plane of the crystal.Here, θ corresponds to the angle between the magnetic field and the c-axis.Despite the presence of step-like signatures, observing the quantum Hall effect in our case proves challenging, primarily due to the considerable thickness of 70 μm.Nevertheless, as a candidate of 3D quantum Hall system, it is possible to observe some signatures such as 3D quantum Hall effect, which still deserve more experimental verification.(b) FFT analysis for the SdH oscillation in (a).Peaks correspond to the oscillations from electrons and holes.(c) Normalized FFT frequency f (θ = 0) / f (θ) as a function of the angle for the electron and hole oscillations, both of which obey a cos(θ)-dependence (solid line).High mobility and carrier concentration.(a) The Hall resistivity of GS under small magnetic fields from -0.3 T to 0.3 T, at different temperatures from 2 to 16 K.The observed nonlinear Hall curve is a characteristic of two-carrier transport, which can be described by the two-carrier model[5]:   =  ℎ  ℎ 2 −    2 )+ ℎ 2   2  2 ( ℎ −  ) ( ℎ  ℎ +    ) 2 + ℎ 2   2  2 ( ℎ −  ) 2  , where ne(nh) and μe(μh) are the carrier density and mobility of electrons (holes), respectively.An approximation (μe≈μh) has been applied at low temperature.The temperature dependence of electron and hole mobilities from 2 to 16 K has been shown in (b).An extremely high mobility  ≈ 3 × 10 6  2  −1  −1 and large carrier concentration  ,ℎ ≈ 2 × 10 19  −3 (shown in (c)) can be observed at 2 K.The carrier concentration exhibits an anomalous increase at low temperatures, correlating with the metal-insulator transition occurring in this temperature regime.